Testable Implications of General Equilibrium Models: An Integer - - PowerPoint PPT Presentation

Introduction Testable implications on the eq manifold Computational complexity IP Application Conclusion

Testable Implications of General Equilibrium Models: An Integer Programming Approach

Laurens Cherchye Thomas Demuynck Bram De Rock

CES, KU Leuven Dauphine Workshop on Economic Theory “Recent Advances in Revealed Preference Theory: testable restrictions in markets and game” November 25-26, 2010 1 / 25 Testable Implications of General Equilibrium Models: An Integer Programming Approach

Introduction Testable implications on the eq manifold Computational complexity IP Application Conclusion

“We present a finite system of polynomial inequalities in unobservable variables and market data that observations on market prices, individual incomes, and aggregate endowments must satisfy to be consistent with the equilibrium behavior of some pure trade economy.” Brown and Matzkin, 1996

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Introduction Testable implications on the eq manifold Computational complexity IP Application Conclusion

“We present a finite system of polynomial inequalities in unobservable variables and market data that observations on market prices, individual incomes, and aggregate endowments must satisfy to be consistent with the equilibrium behavior of some pure trade economy.” “To apply the methodology to large data sets, it is necessary to devise a computationally efficient algorithm for solving large families of equilibrium inequalities.” Brown and Matzkin, 1996

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Sonnenschein-Mantel-Debreu

Consumers with utility functions uj, endowments εj choose consumption given the prices p′.

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Sonnenschein-Mantel-Debreu

Consumers with utility functions uj, endowments εj choose consumption given the prices p′. This gives demand functions xj(p, p′εj).

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Sonnenschein-Mantel-Debreu

Consumers with utility functions uj, endowments εj choose consumption given the prices p′. This gives demand functions xj(p, p′εj). Excess demand function: Zε(p) =

j xj(p, p′εj) − j εj.

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Sonnenschein-Mantel-Debreu

Consumers with utility functions uj, endowments εj choose consumption given the prices p′. This gives demand functions xj(p, p′εj). Excess demand function: Zε(p) =

j xj(p, p′εj) − j εj.

Sonnenschein-Mantel-Debreu: any real valued function (Zε)

• f prices that satisfies Walras’ law, continuity and

homogeneity of degree zero is the excess demand function of some economy with at least as many agents as commodities.

3 / 25 Testable Implications of General Equilibrium Models: An Integer Programming Approach

Introduction Testable implications on the eq manifold Computational complexity IP Application Conclusion

Sonnenschein-Mantel-Debreu

Consumers with utility functions uj, endowments εj choose consumption given the prices p′. This gives demand functions xj(p, p′εj). Excess demand function: Zε(p) =

j xj(p, p′εj) − j εj.

Sonnenschein-Mantel-Debreu: any real valued function (Zε)

• f prices that satisfies Walras’ law, continuity and

homogeneity of degree zero is the excess demand function of some economy with at least as many agents as commodities. “Anything goes!” (Mas-Colell, Whinston and Green, 1995)

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The Equilibrium Manifold

Balasko (2006): What we observe is is not Zε(p), but E = {(ε, p)|Zε(p) = 0}.

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The Equilibrium Manifold

Balasko (2006): What we observe is is not Zε(p), but E = {(ε, p)|Zε(p) = 0}. Brown-Matzkin (1996): What are the necessary and sufficient conditions on a finite data set on (equilibrium) prices, aggregate endowments and individual incomes such that this data is consistent with observations from the equilibrium manifold.

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The Equilibrium Manifold

Balasko (2006): What we observe is is not Zε(p), but E = {(ε, p)|Zε(p) = 0}. Brown-Matzkin (1996): What are the necessary and sufficient conditions on a finite data set on (equilibrium) prices, aggregate endowments and individual incomes such that this data is consistent with observations from the equilibrium manifold. Using Revealed preference theory, they show that not every data set is consistent with equilibrium behavior.

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Proof outline

1

Develop Revealed Preference conditions that guarantee the existence of utility functions and individual consumption bundles such that:

Individual expenditure equals individual income, Individual consumption sums to aggregate endowment, Individual consumption maximizes the individual utility function given the available income.

These conditions form a set of polynomial inequalities.

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Proof outline

1

Develop Revealed Preference conditions that guarantee the existence of utility functions and individual consumption bundles such that:

Individual expenditure equals individual income, Individual consumption sums to aggregate endowment, Individual consumption maximizes the individual utility function given the available income.

These conditions form a set of polynomial inequalities.

2

Employ Tarski-Seidenberg algorithm: Any finite system of polynomial inequalities can be reduced to an equivalent finite family of polynomial inequalities in the coefficients of the given system.

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Proof outline

1

Develop Revealed Preference conditions that guarantee the existence of utility functions and individual consumption bundles such that:

Individual expenditure equals individual income, Individual consumption sums to aggregate endowment, Individual consumption maximizes the individual utility function given the available income.

These conditions form a set of polynomial inequalities.

2

Employ Tarski-Seidenberg algorithm: Any finite system of polynomial inequalities can be reduced to an equivalent finite family of polynomial inequalities in the coefficients of the given system.

3

Provide a counterexample.

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Counterexample

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Other research

Public goods (Snyder, 1999); financial markets (K¨ ubler, 2003); random preferences (Carvajal, 2004); Pareto efficiency (Bachman, 2006); interdependent preferences (Deb, 2009); externalities (Carvajal, 2009),. . .

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Other research

Public goods (Snyder, 1999); financial markets (K¨ ubler, 2003); random preferences (Carvajal, 2004); Pareto efficiency (Bachman, 2006); interdependent preferences (Deb, 2009); externalities (Carvajal, 2009),. . . proof strategy is mostly the same.

1

Derive RP conditions that form a set of polynomial inequalities.

2

Use the Tarski-Seidenberg algorithm.

3

Provide a counterexample.

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Introduction Testable implications on the eq manifold Computational complexity IP Application Conclusion

Other research

Public goods (Snyder, 1999); financial markets (K¨ ubler, 2003); random preferences (Carvajal, 2004); Pareto efficiency (Bachman, 2006); interdependent preferences (Deb, 2009); externalities (Carvajal, 2009),. . . proof strategy is mostly the same.

1

Derive RP conditions that form a set of polynomial inequalities.

2

Use the Tarski-Seidenberg algorithm.

3

Provide a counterexample.

Tarski-Seidenberg: can it be used to operationalize the general equilibrium conditions?

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Tarski-Seidenberg

“It may be difficult, using the TarskiSeidenberg algorithm, to derive these testable restrictions on the equilibrium manifold in a computationally efficient manner for every finite data set.” (Brown and Matzkin, 1996)

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Tarski-Seidenberg

“It may be difficult, using the TarskiSeidenberg algorithm, to derive these testable restrictions on the equilibrium manifold in a computationally efficient manner for every finite data set.” (Brown and Matzkin, 1996) “The algorithm is doubly-exponential in practice, however, and thus not feasible for many problems.”

(Carvajal, Ray and Snyder, 2004)

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Tarski-Seidenberg

“It may be difficult, using the TarskiSeidenberg algorithm, to derive these testable restrictions on the equilibrium manifold in a computationally efficient manner for every finite data set.” (Brown and Matzkin, 1996) “The algorithm is doubly-exponential in practice, however, and thus not feasible for many problems.”

(Carvajal, Ray and Snyder, 2004)

“. . . to apply the method to large data sets, researchers would need an efficient way to solve large systems of nonlinear polynomial inequalities.” (Rizvi, 2006)

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Outline

We show that the Brown-Matzkin conditions are difficult to verify.

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Outline

We show that the Brown-Matzkin conditions are difficult to verify. We present an integer programming approach to verify the

• conditions. widely used approach to model and handle

NP-complete problems.

Widely available. Very flexible in order to analyze alternative models.

9 / 25 Testable Implications of General Equilibrium Models: An Integer Programming Approach

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Outline

We show that the Brown-Matzkin conditions are difficult to verify. We present an integer programming approach to verify the

• conditions. widely used approach to model and handle

NP-complete problems.

Widely available. Very flexible in order to analyze alternative models. Alternative algorithms: Brown and Kannan (2008): Enumerate preference orderings. Brown and Kannan (2008): VC algorithm.

9 / 25 Testable Implications of General Equilibrium Models: An Integer Programming Approach

Introduction Testable implications on the eq manifold Computational complexity IP Application Conclusion

Outline

We show that the Brown-Matzkin conditions are difficult to verify. We present an integer programming approach to verify the

• conditions. widely used approach to model and handle

NP-complete problems.

Widely available. Very flexible in order to analyze alternative models. Alternative algorithms: Brown and Kannan (2008): Enumerate preference orderings. Brown and Kannan (2008): VC algorithm.

We illustrate the IP algorithm using US data.

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Testable implications on the equilibrium manifold

Definition (General equilibrium rationalizability) {pt, εt, I j

t }t∈T is general equilibrium rationalizable if there exist

xj

t and uj(.) such that:

• j xj

t = εt,

p′

txj t = I j t ,

xj

t ∈ arg maxx uj(x) s.t.

p′

tx ≤ I j t .

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Individual rationalizability

Definition (Individual rationalizability) {pt, xt} is individual rationalizable if there exist a utility function u(.) such that:

xt ∈ arg maxx u(x) s.t. p′

tx ≤ It.

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Individual rationalizability

Theorem (Afriat, 1967), (Varian, 1982) A data set {pt, xt}t∈T is individual rationalizable if :

There exist numbers Ut and λt > 0 such that: Ut − Uv ≤ λvp′

v(xt − xv),

{pt, xt}t∈T satisfies GARP.

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Individual rationalizability

Theorem (Afriat, 1967), (Varian, 1982) A data set {pt, xt}t∈T is individual rationalizable if :

There exist numbers Ut and λt > 0 such that: Ut − Uv ≤ λvp′

v(xt − xv),

{pt, xt}t∈T satisfies GARP.

Definition (GARP) There exist a function R such that:

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Individual rationalizability

Theorem (Afriat, 1967), (Varian, 1982) A data set {pt, xt}t∈T is individual rationalizable if :

There exist numbers Ut and λt > 0 such that: Ut − Uv ≤ λvp′

v(xt − xv),

{pt, xt}t∈T satisfies GARP.

Definition (GARP) There exist a function R such that:

if ptxt ≥ p′

txv then xtRxv,

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Individual rationalizability

Theorem (Afriat, 1967), (Varian, 1982) A data set {pt, xt}t∈T is individual rationalizable if :

There exist numbers Ut and λt > 0 such that: Ut − Uv ≤ λvp′

v(xt − xv),

{pt, xt}t∈T satisfies GARP.

Definition (GARP) There exist a function R such that:

if ptxt ≥ p′

txv then xtRxv,

if xtRxvRxk then xtRxk,

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Individual rationalizability

Theorem (Afriat, 1967), (Varian, 1982) A data set {pt, xt}t∈T is individual rationalizable if :

There exist numbers Ut and λt > 0 such that: Ut − Uv ≤ λvp′

v(xt − xv),

{pt, xt}t∈T satisfies GARP.

Definition (GARP) There exist a function R such that:

if ptxt ≥ p′

txv then xtRxv,

if xtRxvRxk then xtRxk, if xtRxv then p′

vxv ≤ pvxt.

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Characterization

Theorem [Brown and Matzkin, 1996] {pt, εt, I j

t }t∈T is general equilibrium rationalizable if there exist

xj

t such that:

• j xj

t = εt,

p′

txj t = I j t ,

and,

there exist Uj

t and λj t > 0 such that:

Uj

t − Uj v ≤ λj vp′ v(xj t − xj v)

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Characterization

Theorem [Brown and Matzkin, 1996] {pt, εt, I j

t }t∈T is general equilibrium rationalizable if there exist

xj

t such that:

• j xj

t = εt,

p′

txj t = I j t ,

and,

there exist Uj

t and λj t > 0 such that:

Uj

t − Uj v ≤ λj vp′ v(xj t − xj v)

• r {pt, xj

t}t∈T satisfies GARP.

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Computational complexity

• j xj

t = ε

p′

txj t = I j t

Uj

t − Uj v ≤ λj vp′ v(xj t − xj v)

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Computational complexity

• j xj

t = ε

linear p′

txj t = I j t

Uj

t − Uj v ≤ λj vp′ v(xj t − xj v)

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Computational complexity

• j xj

t = ε

linear p′

txj t = I j t

linear Uj

t − Uj v ≤ λj vp′ v(xj t − xj v)

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Computational complexity

• j xj

t = ε

linear p′

txj t = I j t

linear Uj

t − Uj v ≤ λj vp′ v(xj t − xj v)

quadratic

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Computational complexity

• j xj

t = ε

linear p′

txj t = I j t

linear Uj

t − Uj v ≤ λj vp′ v(xj t − xj v)

quadratic Question Is it possible to find an efficient way to solve these conditions?

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Computational complexity

• j xj

t = ε

linear p′

txj t = I j t

linear Uj

t − Uj v ≤ λj vp′ v(xj t − xj v)

quadratic Question Is it possible to find an efficient way to solve these conditions? Answer No, unless P = NP

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Computational complexity

• j xj

t = ε

linear p′

txj t = I j t

linear Uj

t − Uj v ≤ λj vp′ v(xj t − xj v)

quadratic Question Is it possible to find an efficient way to solve these conditions? Answer No, unless P = NP : the rationalizability conditions are NP- Complete.

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NP-Completeness

Polynomial time: efficient, (e.g. x2).

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NP-Completeness

Polynomial time: efficient, (e.g. x2). Exponential time: inefficient (e.g. 2x).

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NP-Completeness

Polynomial time: efficient, (e.g. x2). Exponential time: inefficient (e.g. 2x). NP-complete: no efficient algorithm exists (unless P = NP).

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Exponential versus polynomial

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Exponential versus polynomial

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Exponential versus polynomial

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The IP program

Question Can we improve upon the Tarski-Seidenberg algorithm?

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Introduction Testable implications on the eq manifold Computational complexity IP Application Conclusion

The IP program

Question Can we improve upon the Tarski-Seidenberg algorithm? Answer Yes.

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The IP program

Question Can we improve upon the Tarski-Seidenberg algorithm? Answer

• Yes. We propose an Integer Programming approach (IP).

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The IP program

Question Can we improve upon the Tarski-Seidenberg algorithm? Answer

• Yes. We propose an Integer Programming approach (IP).

IP is linear programming where some variables are restricted to take only integer values.

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Introduction Testable implications on the eq manifold Computational complexity IP Application Conclusion

The IP program

Question Can we improve upon the Tarski-Seidenberg algorithm? Answer

• Yes. We propose an Integer Programming approach (IP).

IP is linear programming where some variables are restricted to take only integer values. It is a widely used approach to model and handle NP-complete problems.

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Introduction Testable implications on the eq manifold Computational complexity IP Application Conclusion

The IP program

Question Can we improve upon the Tarski-Seidenberg algorithm? Answer

• Yes. We propose an Integer Programming approach (IP).

IP is linear programming where some variables are restricted to take only integer values. It is a widely used approach to model and handle NP-complete problems. There exist good software packages that solve moderate sized instances of IP problems in reasonable time.

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Introduction Testable implications on the eq manifold Computational complexity IP Application Conclusion

The IP program

Question Can we improve upon the Tarski-Seidenberg algorithm? Answer

• Yes. We propose an Integer Programming approach (IP).

IP is linear programming where some variables are restricted to take only integer values. It is a widely used approach to model and handle NP-complete problems. There exist good software packages that solve moderate sized instances of IP problems in reasonable time. It is also very flexible in order to analyze alternative general equilibrium models.

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The IP approach to GARP

Set r(v, t) = 1 if and only if xtRxv GARP conditions I: if p′

txt ≥ p′ txv then

xtRxv, II: if xtRxvRxk then xtRxk, III: if xtRxv then p′

vxv ≤ p′ vxt,

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The IP approach to GARP

Set r(v, t) = 1 if and only if xtRxv GARP conditions I: if p′

txt ≥ p′ txv then

xtRxv, p′

t(xt − xv) < r(t, v)A

II: if xtRxvRxk then xtRxk, III: if xtRxv then p′

vxv ≤ p′ vxt,

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The IP approach to GARP

Set r(v, t) = 1 if and only if xtRxv GARP conditions I: if p′

txt ≥ p′ txv then

xtRxv, p′

t(xt − xv) < r(t, v)A

II: if xtRxvRxk then xtRxk, r(t, v) + r(v, k) ≤ 1 + r(t, k) III: if xtRxv then p′

vxv ≤ p′ vxt,

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The IP approach to GARP

Set r(v, t) = 1 if and only if xtRxv GARP conditions I: if p′

txt ≥ p′ txv then

xtRxv, p′

t(xt − xv) < r(t, v)A

II: if xtRxvRxk then xtRxk, r(t, v) + r(v, k) ≤ 1 + r(t, k) III: if xtRxv then p′

vxv ≤ p′ vxt,

(1 − r(t, v))A ≥ p′

v(xv − xt)

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The IP program

Exchange Economy {pt, εt, I j

t }t∈T is general equilibrium rationalizable if there exist

xj

t such that:

• j xj

t = εt,

p′

txj t = I j t ,

{pt, xj

t}t∈T satisfies GARP.

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The IP program

Exchange Economy {pt, εt, I j

t }t∈T is general equilibrium rationalizable if there exist

xj

t and r j(t, v) ∈ {0, 1} such that:

• j xj

t = εt,

p′

txj t = I j t ,

p′

t(xj t − xj v) < rj(t, v)A,

rj(t, v) + rj(v, k) ≤ 1 + rj(t, k), (1 − rj(t, v))A ≥ p′

v(xj v − xj t).

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The IP program

Lower bound on income {pt, εt, I j

t }t∈T is general equilibrium rationalizable if there exist

xj

t such that:

• j xj

t = εt,

p′

txj t ≥ I j t ,

p′

t(xj t − xj v) < rj(t, v)A,

rj(t, v) + rj(v, k) ≤ 1 + rj(t, k), (1 − rj(t, v))A ≥ p′

v(xj v − xj t).

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The IP program

Assignable information on consumption {pt, εt, I j

t }t∈T is general equilibrium rationalizable if there exist

xj

t such that:

• j xj

t = εt,

p′

txj t = I j t ,

p′

t(xj t − xj v) < rj(t, v)A,

rj(t, v) + rj(v, k) ≤ 1 + rj(t, k), (1 − rj(t, v))A ≥ p′

v(xj v − xj t).

xj

t ≥ ¯

xj

t

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The IP program

Pareto provision of public goods {pt, εt, I j

t }t∈T is general equilibrium rationalizable if there exist

xj

t, Pj t and r j(t, v) such that:

• j xj

t = εt,

p′

txj t + Pj′ t Qt = I j t ,

p′

t(xj t − xj v) + Pj′ t (Qt − Qv) < rj(t, v)I j t ,

rj(t, v) + rj(v, k) ≤ 1 + rj(t, k), (1 − rj(t, v))I j

v ≥ p′ v(xj v − xj t) + Pj′ v (Qv − Qt).

• j Pj

t = Pt.

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The IP program

Private provision of public goods {pt, εt, I j

t }t∈T is general equilibrium rationalizable if there exist

xj

t, Pj t and r j(t, v) such that:

• j xj

t = εt,

p′

txj t + Pj′ t Qt = I j t ,

p′

t(xj t − xj v) + Pj′ t (Qt − Qv) < rj(t, v)I j t ,

rj(t, v) + rj(v, k) ≤ 1 + rj(t, k), (1 − rj(t, v))I j

v ≥ p′ v(xj v − xj t) + Pj′ v (Qv − Qt).

maxj{Pj

t} = Pt.

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Application

US aggregate data. T: 12 observations: 1997-2008 εt: 18 goods I j

t : national incomes for 51 states or 8 regions.

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Application

US aggregate data. T: 12 observations: 1997-2008 εt: 18 goods I j

t : national incomes for 51 states or 8 regions.

IP test for 51 states: pass after 19 minutes.

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Application

US aggregate data. T: 12 observations: 1997-2008 εt: 18 goods I j

t : national incomes for 51 states or 8 regions.

IP test for 51 states: pass after 19 minutes. We choose the 8 regions for power analysis.

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Power analysis

Assignable information: ¯ xj

t =

I i

t

ptεt εt. Require that xj

t ≥ κ¯

xj

• t. (κ ∈ [0, 1])

The lower κ, the less assignable information. Alternative scenario: defense is a public good.

22 / 25 Testable Implications of General Equilibrium Models: An Integer Programming Approach

Introduction Testable implications on the eq manifold Computational complexity IP Application Conclusion

Power Results

Test Power for basic model Power for model with public consumption κ = 1.0 0.6372 0.5934 κ = 0.9 0.2946 0.0479 κ = 0.8 0.1211 0.0084 κ = 0.7 0.0544 0.0034 κ = 0.6 0.0333 0.0016 κ = 0.5 0.0180 0.0008 κ = 0.4 0.0135 0.0003 κ = 0.3 0.0088 0.0001 κ = 0.2 0.0064 0.0001 κ = 0.1 0.0032 0.0001 κ = 0.0 0.0000 0.0000

23 / 25 Testable Implications of General Equilibrium Models: An Integer Programming Approach

Introduction Testable implications on the eq manifold Computational complexity IP Application Conclusion

Power Results

24 / 25 Testable Implications of General Equilibrium Models: An Integer Programming Approach

Introduction Testable implications on the eq manifold Computational complexity IP Application Conclusion

Conclusion

Testable restrictions on equilibrium manifold are difficult to verify: NP-Complete.

25 / 25 Testable Implications of General Equilibrium Models: An Integer Programming Approach

Introduction Testable implications on the eq manifold Computational complexity IP Application Conclusion

Conclusion

Testable restrictions on equilibrium manifold are difficult to verify: NP-Complete. MIP allows easily implementable necessary and sufficient conditions.

25 / 25 Testable Implications of General Equilibrium Models: An Integer Programming Approach

Introduction Testable implications on the eq manifold Computational complexity IP Application Conclusion

Conclusion

Testable restrictions on equilibrium manifold are difficult to verify: NP-Complete. MIP allows easily implementable necessary and sufficient conditions. The approach is flexible to consider extension towards other

• gen. eq. models (e.g. with public goods).

25 / 25 Testable Implications of General Equilibrium Models: An Integer Programming Approach

Introduction Testable implications on the eq manifold Computational complexity IP Application Conclusion

Conclusion

Testable restrictions on equilibrium manifold are difficult to verify: NP-Complete. MIP allows easily implementable necessary and sufficient conditions. The approach is flexible to consider extension towards other

• gen. eq. models (e.g. with public goods).

Assignable info is important to increase power.

25 / 25 Testable Implications of General Equilibrium Models: An Integer Programming Approach

Introduction Testable implications on the eq manifold Computational complexity IP Application Conclusion

Conclusion

Testable restrictions on equilibrium manifold are difficult to verify: NP-Complete. MIP allows easily implementable necessary and sufficient conditions. The approach is flexible to consider extension towards other

• gen. eq. models (e.g. with public goods).

Assignable info is important to increase power. Future topics:

heuristics, special cases that are efficiently verifiable (e.g. quasi-linear utility) recovery, goodness of fit.

25 / 25 Testable Implications of General Equilibrium Models: An Integer Programming Approach